Graded Geometry, $Q$-Manifolds, and Microformal Geometry

Abstract

We give an exposition of graded and microformal geometry, and the language of $Q$-manifolds. $Q$-manifolds are supermanifolds endowed with an odd vector field of square zero. They can be seen as a non-linear analogue of Lie algebras (in parallel with even and odd Poisson manifolds), a basis of "non-linear homological algebra", and a powerful tool for describing algebraic and geometric structures. This language goes together with that of graded manifolds, which are supermanifolds with an extra $\mathbb{Z}$-grading in the structure sheaf. "Microformal geometry" is a new notion referring to "thick" or "microformal" morphisms, which generalize ordinary smooth maps, but whose crucial feature is that the corresponding pullbacks of functions are nonlinear. In particular, "Poisson thick morphisms" of homotopy Poisson supermanifolds induce $L_{\infty }$-morphisms of homotopy Poisson brackets. There is a quantum version based on special type Fourier integral operators and applicable to Batalin-Vilkovisky geometry. Though the text is mainly expository, some results are new or not published previously.

34 pages, Contribution to Proceedings of LMS/EPSRC Durham Symposium Higher Structures in M-Theory, August 2018

References in corpus (8)

• Deformation theory and rational homotopy type
• $Q$-manifolds and Higher Analogs of Lie Algebroids
• $Q$-Manifolds and Mackenzie Theory
• Homological Algebra for Superalgebras of Differentiable Functions
• Q-manifolds and Mackenzie theory: an overview
• Formal oscillatory integrals and deformation quantization
• Thick morphisms, higher Koszul brackets, and $L_{\infty }$-algebroids
• Tangent functor on microformal morphisms, and non-linear pullbacks for forms and cohomology

Cited by in corpus (9)

• Global Theory of Graded Manifolds
• Higher Spin Gravities and Presymplectic AKSZ Models
• Dg manifolds, formal exponential maps and homotopy Lie algebras
• Double Principal Bundles
• L-infinity bialgebroids and homotopy Poisson structures on supermanifolds
• Tangent functor on microformal morphisms, and non-linear pullbacks for forms and cohomology
• Normal forms of $\mathbb{Z}$-graded $Q$-manifolds
• Thick morphisms of supermanifolds, quantum mechanics, and spinor representation
• Threefold Nature of Graded Vector Bundles